The best known lower bounds on the field size required for \((n,r,h,a,g)\) -MR LRCs (with \(gr=n\) ) is from [1]} who show that for \(h\ge 2\) ,
\(q \ge \Omega _{h,a} \left( n\cdot r^\alpha \right) \text{ where } \alpha =\frac{\min \left\lbrace a,h-2\lceil h/g\rceil \right\rbrace }{\lceil h/g \rceil }.\) 
